In the well-known Doppler equation, the frequency observed by a moving platform is shifted in a nonlinear fashion relative to the emitter's carrier frequency fe and the geometric relationship between the emitter and the platform, as shown below in Equation 1:
                              Δ          ⁢                                          ⁢          f                =                              (                                          v                ⁢                                                                  ⁢                cos                ⁢                                                                  ⁢                θ                            c                        )                    ⁢                      f            e                                              Equation        ⁢                                  ⁢        1            
where Δf is the Doppler shift, v is the platform speed, θ is the cone angle between the emitter and the platform velocity vector, and c is the speed of light. Multiple Doppler observations can be triangulated and used to solve for the unknown emitter location.
To date, the most common methods for geolocation of uncooperative RF emitters employ multilateration (i.e., multiple platforms). Typically these methods employ measurements of Time Difference of Arrival (“TDOA”) and/or Frequency Difference of Arrival (“FDOA”) of radar signals using at least two sensors located on separate platforms. For example, the 3D position of an emitter may be determined by measuring the TDOA of a signal at four or more spatially separate platform receiver sites located different distances from the signal source. If the platform and emitter are in relative motion with respect to one another, FDOA may be employed either independently or in addition to TDOA to determine the emitter position based on vector velocities and the observed relative Doppler shift.
FDOA uses the difference of frequencies collected simultaneously by at least two different moving sensors. As shown in Equation 2, below, the FDOA observable Δω describes the Doppler shift caused by the relative change in range rate between two collection platforms:
                              Δ          ⁢                                          ⁢          ω                =                                            f              e                        c                    ⁢                      ⅆ                          ⅆ              t                                ⁢                      (                                          r                1                            -                              r                2                                      )                                              Equation        ⁢                                  ⁢        2            where the variable rk represents the range between a receiver k and the emitter.
A primary advantage of the FDOA observable is that it largely removes the geolocation algorithm's dependence on an unknown emitter's frequency. This insensitivity is seen in the Equation 2 where the unknown emitter frequency fe is divided by the speed of light. Because the emitter's frequency is often in the hundreds of MHz and the speed of light is roughly 3×108 m/s, large uncertainty in the emitter frequency estimate is usually still well below the sensor noise floor. This almost invariance to emitter frequency uncertainty greatly simplifies the geolocation solution by allowing the state space to be composed of only the emitter position. This stationary state space (no physical emitter movement) can be solved efficiently using, for example, an iterative Gauss-Newton gradient descent algorithm.
Unfortunately, there are a number of disadvantage to employing FDOA-based emitter location algorithms in practice, such as the requirement for two simultaneous platforms. Furthermore, local oscillators (sensors) onboard small unmanned aerial vehicles (UAVs) that baseband received signals are often relatively stable, but biased. Because the relative difference of these two local oscillators forms the basis of the FDOA measurement, it is important that the biases are corrected. This is sometimes done using known calibration tones present in the collected data, however, this requires even more resources and further motivates a robust, single sensor, FOA geolocation solution.
The FOA measurement equation is derived directly from Equation 1 by rearranging terms to isolate the observed Doppler-shifted frequency fobs from the unobserved Doppler shift Δf that depends on the emitter's unknown carrier frequency fe. Furthermore, the cosine term can be expanded to explicitly contain the state variables of emitter location [xe,ye]T and emitter frequency fe by writing out the dot product between the sensor velocity [vx,vy]T and the line of sight between the emitter and the collection platform [x,y]T:
                              f          obs                =                              [                          1              -                                                1                  c                                ⁢                                                                                                    (                                                  x                          -                                                      x                            e                                                                          )                                            ⁢                                              v                        x                                                              +                                                                  (                                                  y                          -                                                      y                            e                                                                          )                                            ⁢                                              v                        y                                                                                                  c                    ⁢                                                                                                                        (                                                          x                              -                                                              x                                e                                                                                      )                                                    2                                                +                                                                              (                                                          y                              -                                                              y                                e                                                                                      )                                                    2                                                                                                                                          ]                    ⁢                      f            e                                              Equation        ⁢                                  ⁢        3            
A primary reason that the FOA method is generally not considered an operational geolocation option is due to its tendency to produce unreliable results. Without a priori knowledge of an emitter's transmission frequency, a geolocation engine must simultaneously solve for ground location and frequency of the emitter of interest. Provided a stable carrier, the Gauss-Newton method, among others, may product an adequate solution over the course of a collection long enough to obtain sufficient measurement independence for a unique solution. However, the Gauss-Newton method has a strong tendency to produce erroneous estimates if insufficient geometries are collected due to the correspondence of many possible solutions to the observations. In these cases, the optimization often gets trapped in erroneous local minima. In other words, the expected solution may converge to one of several potential solutions that are false.
The situation is made more challenging by the fact that most commercial transmitter systems do not have strict tolerances on oscillator stability which means that an emitter's carrier frequency can drift significantly during normal operation. Failing to account for frequency drift observed over the course of triangulation time required to determine a unique solution will cause gross inaccuracies in geolocation estimates. However, including a non-stationary drift term in the geolocation solution eliminates the suitability of Gauss Newton and similar techniques, because it requires a stationary state model. This motivates the need for a new solution that can efficiently represent and track in a non-linear, non-convex, multimodal state space.